Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{x^2 - 25}{x - 5}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{25} = -5$ So we can rewrite the expression as: $r = \dfrac{({x} {-5})({x} + {5})} {x - 5} $ We can divide the numerator and denominator by $(x - 5)$ on condition that $x \neq 5$ Therefore $r = x + 5; x \neq 5$